Volume of Revolution: The Disk Method
Simple Explanation
Rotating the region under a curve f(x) around the x-axis sweeps out a solid of revolution. Slicing that solid perpendicular to the axis produces thin circular disks of radius f(x) β summing their volumes via integration gives V = Οβ«βα΅[f(x)]Β²dx.
Why Do We Need It?
Many real solids β bottles, vases, machine parts turned on a lathe β are naturally described as a curve rotated around an axis, and the disk method is the direct way to compute their volume.
See It
A leaf-shaped side-view silhouette symmetric about a horizontal axis, representing the outline of a solid of revolution, with a dashed vertical line showing one disk's diameter
Formula
Volume of Revolution: The Disk Method
V = Ο β«βα΅ [f(x)]Β² dx
Rotating the region under a curve f(x) around the x-axis sweeps out a solid; slicing it into thin circular disks of radius f(x) and summing their volumes (via integration) gives the total volume.
- f(x)
- β the radius of the disk at position x β the distance from the x-axis to the curve
- a, b
- β the x-boundaries of the region being revolved
When to use it: Whenever the region under a single curve (with no gap between the curve and the axis of rotation) is revolved around that axis.
Worked Example
Find a volume of revolution using the disk method
Find the volume generated by rotating the region under y=βx, from x=0 to x=4, around the x-axis.
Why Does This Work?
Each thin disk at position x has radius f(x) and thickness dx, so its volume is Ο[f(x)]Β²Β·dx (the area of a circle, times a thin slice of thickness) β summing infinitely many such disks across [a,b] via integration gives the total volume of the whole solid.
Real-Life Example
Computing the volume of a lathe-turned bottle
A glassblower or machinist designs a bottle's profile as a curve, then spins it on a lathe (or the equivalent process) to create the full three-dimensional shape.
The bottle's volume can be computed in advance directly from its profile curve using the disk method, without needing to fill a physical prototype with liquid to measure it.
Practice
Find the volume when the region under y=x, from x=0 to x=2, is rotated around the x-axis.
HardCommon mistake
Forgetting to square f(x) before integrating β the disk method needs [f(x)]Β² (since a disk's area is ΟrΒ², not Οr), so integrating f(x) alone gives a completely wrong result.
Quick Review
- V = Οβ«βα΅ [f(x)]Β² dx β for a region under f(x), rotated about the x-axis.
- Each disk has area Ο[f(x)]Β² and infinitesimal thickness dx.
- Only valid when the region touches the axis of rotation directly (no gap).